Shunya / Mensuration & 3D Shapes Lab

Shunya · Maths lab

Mensuration & 3D shapes lab

Measure flat and solid shapes. Drag the corners of a triangle and check Heron's formula, turn cuboids, cylinders, cones, spheres and frustums in 3D while their surface area and volume update, unfold each solid into its net, stack solids together, and melt one solid to see how many smaller ones it makes.

Lessons

Pick your class, follow the steps in the lab, and check what you see against the working under the graph.

Class 9 — Area of a triangle and Heron's formula

Concept: the area of a triangle is ½ × base × height. When you only know the three sides a, b, c, Heron's formula gives it: with s = (a + b + c) ÷ 2, the area is √(s(s − a)(s − b)(s − c)).

Open Plane figures, choose Triangle (Heron) and press 3-4-5 right triangle: the area is 6.
Read the working: s = 6, then √(6 × 3 × 2 × 1) = 6, and check ½ × 4 × 3 = 6.
Drag a corner to make any triangle and compare the three area cards: Heron, ½ × base × height and the coordinate method always agree.
Press long and thin and see how a very flat triangle has a tiny area.
Drag one corner onto the line through the other two: the area becomes 0 and it is not a triangle.

Takeaway: Heron's formula needs only the three sides, and every method gives the same area.

Class 9 — Parallelograms, trapezia and rhombuses

Concept: a parallelogram has area base × height (the perpendicular height, not the slanted side). A trapezium has area ½ × (sum of parallel sides) × height. A rhombus has area ½ × d₁ × d₂ because its diagonals cross at right angles.

Choose Parallelogram and change the angle: the height changes, the base does not, so the area changes.
Set the angle to 90° and see the rectangle: the area is the same b × h.
Choose Trapezium and slide the top sideways: the area does not change, because the height and the parallel sides are the same.
Read the midline card: area = midline × height.
Choose Rhombus, drag the two diagonals and check the side and perimeter.

Takeaway: always use the perpendicular height, and remember the trapezium's ½ (a + b) h.

Class 9 — Cuboid and cube

Concept: a cuboid has volume l × b × h, lateral surface area 2h(l + b) and total surface area 2(lb + bh + hl). A cube is a cuboid with all edges equal: V = a³, TSA = 6a².

Open 3D solids, choose Cuboid and drag the solid to turn it.
Change l, b and h and watch V and the surface areas update.
Tick the measurements to see which edges are l, b and h.
Choose Cube and check that the total surface area is 6a².
Check the space diagonal √(l² + b² + h²) for a 3 × 4 × 12 box: it is 13.

Takeaway: volume is the layers times the base; surface area is the six faces added.

Class 9 — Cylinder, cone and their curved surfaces

Concept: a cylinder has curved surface 2πrh, total surface 2πr(r + h) and volume πr²h. A cone has slant height l = √(r² + h²), curved surface πrl, total surface πr(r + l) and volume ⅓πr²h.

Choose Cylinder and press can: r = 3, h = 10.
Open Nets & surface area and choose Cylinder: the curved surface unrolls into a rectangle 2πr wide.
Choose Cone in the nets: the curved surface unrolls into a sector whose arc is 2πr.
Read the sector angle 360° × r ÷ l and the area πrl.
In 3D solids choose Cone and press cone r = 3, h = 4 (l = 5).

Takeaway: a curved surface unrolls flat, and its area is the area of the net.

Class 9 — Sphere, hemisphere and the cone–cylinder relation

Concept: a sphere has surface area 4πr² and volume 4/3 πr³. A hemisphere has curved surface 2πr², total surface 3πr² and volume ⅔πr³. A cone is exactly one third of the cylinder with the same base and height.

Choose Sphere and press ball r = 4. Double r and see the volume grow 8 times and the area 4 times.
Choose Hemisphere and compare its total surface 3πr² with half the sphere's.
Open Cone · sphere · cylinder and slide the pour control: three cones fill the cylinder.
Switch to Archimedes and pour the cone and the sphere into the cylinder: it fills exactly (1 : 2 : 3).
Change r and check the ratios do not change.

Takeaway: V(cone) : V(sphere) : V(cylinder) = 1 : 2 : 3 when they share radius r and height 2r.

Class 10 — Combinations of solids

Concept: to find the volume of a combined solid, add the volumes of the parts. For surface area, add only the surfaces that are still outside: the faces where the solids are joined are hidden.

Open Combine & recast and choose the ice-cream cone: a cone with a hemisphere on top.
Read the working: volume = ⅓πr²h + ⅔πr³; surface = πrl + 2πr² (the flat faces where they meet are not counted).
Choose the capsule and check that the two hemispheres make one sphere.
Choose the tent and see that it has no floor and no top disc.
Drag the solid to look at it from all sides.

Takeaway: volumes add; surfaces add only where you can see them.

Class 10 — Melting and recasting solids

Concept: when a solid is melted and made into other solids, the volume stays the same. So the number of small solids is (volume of the big one) ÷ (volume of a small one). The total surface area does change.

Choose Melt and recast and press A sphere r = 6 into spheres r = 2: 27 spheres.
Check: 4/3 π 6³ ÷ 4/3 π 2³ = 216 ÷ 8 = 27.
Try A cylinder into coins and A sphere into cones.
Compare the surface area before and after: it grows.
Change the new solid's size until the answer is a whole number.

Takeaway: recasting keeps the volume; divide volumes to count the new solids.

Class 10 — Frustum of a cone

Concept: a frustum is a cone with its top cut off by a plane parallel to the base. With radii R and r, height h and slant height l = √(h² + (R − r)²): volume ⅓πh(R² + Rr + r²), curved surface π(R + r)l, total surface π(R + r)l + πR² + πr².

Open 3D solids, choose Frustum of a cone and press bucket (frustum).
Make r almost equal to R: it becomes a cylinder. Make r small: it becomes a cone.
Open Nets and choose the frustum: the curved surface is a piece of a ring (an annular sector).
Check that the pieces add up to the total surface area.
Verify the volume as big cone − small cone.

Takeaway: a frustum is a big cone minus a small cone, and it links the cylinder and the cone.

The maths behind it

What each mode covers

Plane figures — a triangle you can drag with Heron's formula checked against ½ × base × height and the coordinate method, parallelograms, trapezia, rhombuses, regular polygons and rings. Class 9.

3D solids — cuboid, cube, cylinder, cone, sphere, hemisphere, frustum and pyramid that you can turn with your finger or mouse, with sliders for every measurement and live volume, curved and total surface area, slant height and diagonal. Classes 9 and 10.

Nets & surface area — the solid cut open and laid flat: the cylinder's rectangle, the cone's sector, the frustum's ring piece, the cuboid's cross. The pieces add up to the total surface area. Classes 9 and 10.

Combine & recast — ice-cream cone, capsule, tent and toy shown in 3D with their volume and visible surface, and a melt-and-recast calculator that counts how many small solids one big solid makes. Class 10.

Cone · sphere · cylinder — pour a cone into a cylinder three times, or a cone and a sphere into a cylinder (Archimedes' 1 : 2 : 3). Class 9.

Part of Shunya. See the syllabus map for the chapters these modes map to. The 3D views are drawn with a small built-in renderer: round solids are shown as many flat pieces, so their outline is very slightly rough, but every number uses the exact formulas. Circles and sectors are in Circles & Conic Sections.

Formulas used in this lab

Area of a triangle Class 9

Base and height: $A = \tfrac{1}{2}\,b\,h$
Semi-perimeter: $s = \dfrac{a + b + c}{2}$
Heron's formula: $A = \sqrt{s(s - a)(s - b)(s - c)}$
Equilateral triangle of side a: $A = \dfrac{\sqrt{3}}{4}\,a^2$
Isosceles (equal sides a, base b): $A = \dfrac{b}{4}\sqrt{4a^2 - b^2}$

Quadrilaterals Class 9

Rectangle and square: $A = lb,\qquad A = a^2$
Parallelogram: $A = b \times h$
Trapezium: $A = \tfrac{1}{2}(a + b)\,h$
Rhombus: $A = \tfrac{1}{2}\,d_1 d_2,\qquad \text{side} = \sqrt{\left(\tfrac{d_1}{2}\right)^2 + \left(\tfrac{d_2}{2}\right)^2}$
Regular polygon: $A = \tfrac{1}{2} \times \text{perimeter} \times \text{apothem}$

Cuboid and cube Class 9

Volume of a cuboid: $V = l \times b \times h$
Lateral surface area: $2h(l + b)$
Total surface area: $2(lb + bh + hl)$
Space diagonal: $\sqrt{l^2 + b^2 + h^2}$
Cube of edge a: $V = a^3,\quad \text{TSA} = 6a^2,\quad d = a\sqrt{3}$

Cylinder Class 9

Volume: $V = \pi r^2 h$
Curved surface area: $\text{CSA} = 2\pi r h$
Total surface area: $\text{TSA} = 2\pi r(r + h)$
Hollow cylinder: $V = \pi h\,(R^2 - r^2)$

Cone Class 9

Slant height: $l = \sqrt{r^2 + h^2}$
Volume: $V = \tfrac{1}{3}\pi r^2 h$
Curved surface area: $\text{CSA} = \pi r l$
Total surface area: $\text{TSA} = \pi r(r + l)$
The unrolled sector: $\theta = \dfrac{360^\circ \times r}{l}$

Sphere and hemisphere Class 9

Sphere: $V = \tfrac{4}{3}\pi r^3,\qquad S = 4\pi r^2$
Hemisphere volume: $V = \tfrac{2}{3}\pi r^3$
Hemisphere surface areas: $\text{CSA} = 2\pi r^2,\qquad \text{TSA} = 3\pi r^2$
Spherical shell: $V = \tfrac{4}{3}\pi(R^3 - r^3)$
Cone : sphere : cylinder (r, 2r): $1 : 2 : 3$

Combinations of solids Class 10

Volume: $V = V_1 + V_2 + \dots$
Surface area: $\text{visible surfaces only}$
Ice-cream cone (cone + hemisphere): $V = \tfrac{1}{3}\pi r^2 h + \tfrac{2}{3}\pi r^3,\quad S = \pi r l + 2\pi r^2$
Capsule (cylinder + 2 hemispheres): $V = \pi r^2 h + \tfrac{4}{3}\pi r^3,\quad S = 2\pi r h + 4\pi r^2$

Converting one solid into another Class 10

Volume is unchanged: $V_{\text{old}} = V_{\text{new}}$
Number of small solids: $n = \dfrac{V_{\text{big}}}{V_{\text{small}}}$
Sphere into spheres: $n = \left(\dfrac{R}{r}\right)^3$
Surface area: $\text{changes: } n \text{ small solids have more}$

Frustum of a cone Class 10

Slant height: $l = \sqrt{h^2 + (R - r)^2}$
Volume: $V = \tfrac{1}{3}\pi h\,(R^2 + Rr + r^2)$
Curved surface area: $\text{CSA} = \pi(R + r)\,l$
Total surface area: $\text{TSA} = \pi(R + r)\,l + \pi R^2 + \pi r^2$
Big cone minus small cone: $V = \tfrac{1}{3}\pi R^2 H - \tfrac{1}{3}\pi r^2 (H - h)$